Chapter 1.9, Problem 52E

(a)

To determine

To find: inverse of the function of f .

Answer to Problem 52E

The inverse of the given function is $f^{-1}(x)=-\frac{2}{x}$ .

Explanation of Solution

Given information:

The function $f(x)=-\frac{2}{x}$

Calculation:

Let the function $y=f(x)=-\frac{2}{x}$

  $\begin{array}{l}y=-\frac{2}{x}\\ x=-\frac{2}{y}\end{array}$

  $\begin{array}{l}{f}^{-1}(y)=-\frac{2}{y}\\ f^{-1}(x)=-\frac{2}{x}\end{array}$

Hence, the inverse of the given function is $f^{-1}(x)=-\frac{2}{x}$ .

(b)

To determine

To sketch: the graphs both the functions f and $f^{-1}.$

Explanation of Solution

Given information:

The function $f(x)=-\frac{2}{x}$

Graph:

The graph of the functions $f(x)=f^{-1}(x)=-\frac{2}{x}$ are drawn below.

(c)

To determine

To describe: the relation between the graphs of f and $f^{-1}.$

Explanation of Solution

Given information:

The function $f(x)=-\frac{2}{x}$

Consider the graph of part $(b)$

The graph of the $f^{-1}$ is the reflection of graph f about the line $y=x.$

The functions f and $f^{-1}$ both are equal.

(d)

To determine

To state: The domains and ranges of the functions f and $f^{-1}$ .

Explanation of Solution

Given information:

The function $f(x)=-\frac{2}{x}$

Domain is the set of input values for which the function is defined.

Range is the set of defined output values.

From the graphs of part b,

Consider $f(x)=f^{-1}(x)=-\frac{2}{x}$ is

Domain: $-\infty <x<0,0<x<\infty$ , Range: $-\infty <y<0,0<y<\infty$

Hence, the domain of f and $f^{-1}$ are $(-\infty ,0),(0,\infty )$ and range of the functions f and $f^{-1}$

are $(-\infty ,0),(0,\infty )$ respectively.